How to round the edges of this object?
Unfortunately I believe you have started on the wrong foot.

You object has a very bad topology, it's not quad based, it is full of triangles. very unevenly subdivided with extremely high density on top contrasting with huge ngons on the sides, and unnecessarily subdivided in width direction.

All these things make harder to work with, Blender operators will misbehave, that is why you encounter these problems and are likely to expect further down the road.

Modelling applications generally like four sided faces, my advice is to start over with a better mesh base and work from there.



Watch some beginner tutorials on hard-edge modeling and follow these guides for topology